A GapETHTight Approximation Scheme for Euclidean TSP
Abstract
We revisit the classic task of finding the shortest tour of $n$ points in $d$dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{O(1/\varepsilon^{d1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{O(1/\varepsilon^{d1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d1})}\mathrm{poly}(n)$ algorithm would violate the GapExponential Time Hypothesis (GapETH). Our new algorithm builds upon the celebrated quadtreebased methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsitysensitive patching. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. We demonstrate that our technique extends to other problems, by showing that for Steiner Tree and Rectilinear Steiner Tree it yields the same running time. We complement our results with a matching GapETH lower bound for Rectilinear Steiner Tree.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.03778
 Bibcode:
 2020arXiv201103778K
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 43 pages, faster algorithms for Euclidean and Rectilinear Steiner Tree